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12
THE CLASSIFICATION OF pCOMPACT GROUPS FOR p ODD
, 2003
"... A pcompact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined plocal analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to the classification of compact Lie groups. In this paper we fi ..."
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Cited by 28 (14 self)
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A pcompact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined plocal analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to the classification of compact Lie groups. In this paper we finish the proof of this conjecture, for p an odd prime, proving that there is a onetoone correspondence between connected pcompact groups and finite reflection groups over the padic integers. We do this by providing the last, and rather intricate, piece, namely that the exceptional compact Lie groups are uniquely determined as pcompact groups by their Weyl groups seen as finite reflection groups over the padic integers. Our approach in fact gives a largely selfcontained proof of the entire
The classification of 2compact groups
"... Abstract. We prove that any connected 2compact group is classified by its 2adic root datum, and in particular the exotic 2compact group DI(4), constructed by DwyerWilkerson, is the only simple 2compact group not arising as the 2completion of a compact connected Lie group. Combined with our ear ..."
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Cited by 15 (4 self)
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Abstract. We prove that any connected 2compact group is classified by its 2adic root datum, and in particular the exotic 2compact group DI(4), constructed by DwyerWilkerson, is the only simple 2compact group not arising as the 2completion of a compact connected Lie group. Combined with our earlier work with Møller and Viruel for p odd, this establishes the full classification of pcompact groups, stating that, up to isomorphism, there is a onetoone correspondence between connected pcompact groups and root data over the padic integers. As a consequence we prove the maximal torus conjecture, giving a onetoone correspondence between compact Lie groups and finite loop spaces admitting a maximal torus. Our proof is a general induction on the dimension of the group, which works for all primes. It refines the AndersenGrodalMøllerViruel methods to incorporate the theory of root data over the padic integers, as developed by DwyerWilkerson and the authors, and we show that certain occurring obstructions vanish, by relating them to obstruction groups calculated by JackowskiMcClureOliver in the early 1990s. 1.
Finite Chevalley versions of pcompact groups
"... Abstract. We describe the spaces of homotopy xed points of unstable Adams operations acting on pcompact groups and also of unstable Adams operations twisted with a nite order automorphism of the pcompact group. We obtain new exotic plocal nite groups. Contents ..."
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Abstract. We describe the spaces of homotopy xed points of unstable Adams operations acting on pcompact groups and also of unstable Adams operations twisted with a nite order automorphism of the pcompact group. We obtain new exotic plocal nite groups. Contents
A FINITE LOOP SPACE NOT RATIONALLY EQUIVALENT TO A COMPACT LIE GROUP
, 2003
"... Abstract. We construct a connected finite loop space of rank 66 and dimension 1254 whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact Lie group, hence providing a counterexample to a classical conjecture. Aided by machine calculation we ver ..."
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Cited by 6 (2 self)
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Abstract. We construct a connected finite loop space of rank 66 and dimension 1254 whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact Lie group, hence providing a counterexample to a classical conjecture. Aided by machine calculation we verify that our counterexample is minimal, i.e., that any finite loop space of rank less than 66 is in fact rationally equivalent to a compact Lie group, extending the classical known bound of 5. 1.
The Steenrod problem of realizing polynomial cohomology
"... Abstract. In this paper we completely classify which graded polynomial R–algebras in finitely many even degree variables can occur as the singular cohomology of a space with coefficients in R, a 1960 question of N. E. Steenrod, for a commutative ring R satisfying mild conditions. In the fundamental ..."
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Abstract. In this paper we completely classify which graded polynomial R–algebras in finitely many even degree variables can occur as the singular cohomology of a space with coefficients in R, a 1960 question of N. E. Steenrod, for a commutative ring R satisfying mild conditions. In the fundamental case R = Z, our result states that the only polynomial cohomology rings over Z which can occur, are tensor products of copies of H ∗ (CP ∞ ; Z) ∼ = Z[x2], H ∗ (B SU(n); Z) ∼ = Z[x4, x6,..., x2n], and H ∗ (B Sp(n); Z) ∼ = Z[x4, x8,..., x4n], confirming an old conjecture. Our classification extends Notbohm’s solution for R = Fp, p odd. Odd degree generators, excluded above, only occur if R is an F2–algebra and in that case the recent classification of 2–compact groups by the authors can be used instead of the present paper. Our proofs are short and rely on the general theory of p–compact groups, but not on classification results for these. 1.
Automorphisms of pcompact groups and their root data
"... Abstract. We construct a model for the space of automorphisms of a connected pcompact group in terms of the space of automorphisms of its maximal torus normalizer and its root datum. As a consequence we show that any homomorphism to the outer automorphism group of a pcompact group can be lifted to ..."
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Abstract. We construct a model for the space of automorphisms of a connected pcompact group in terms of the space of automorphisms of its maximal torus normalizer and its root datum. As a consequence we show that any homomorphism to the outer automorphism group of a pcompact group can be lifted to a group action, analogous to a classical theorem of de Siebenthal for compact Lie groups. The model of this paper is used in a crucial way in our paper “The classification of 2compact groups”, where we prove the conjectured classification of 2compact groups and determine their automorphism spaces. 1.
Symplectic groups are Ndetermined 2compact groups
, 2008
"... We show that for n ≥ 3 the symplectic group Sp(n) is as a 2compact group determined up to isomorphism by the isomorphism type of its maximal torus normalizer. This allows us to determine the integral homotopy type of Sp(n) among connected finite loop spaces with maximal torus. ..."
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We show that for n ≥ 3 the symplectic group Sp(n) is as a 2compact group determined up to isomorphism by the isomorphism type of its maximal torus normalizer. This allows us to determine the integral homotopy type of Sp(n) among connected finite loop spaces with maximal torus.
Ndetermined 2compact groups
, 2005
"... Key words and phrases. Classification of pcompact groups at the prime p = 2, ..."
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Key words and phrases. Classification of pcompact groups at the prime p = 2,
The Steenrod problem of classifying polynomial cohomology rings
 In preparation
"... Abstract. In this paper we completely classify which graded polynomial Ralgebras in finitely many even degree variables can occur as the singular cohomology of a space with coefficients in R, a 1960 question of N. E. Steenrod, for a commutative ring R satisfying mild conditions. In the fundamental ..."
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Abstract. In this paper we completely classify which graded polynomial Ralgebras in finitely many even degree variables can occur as the singular cohomology of a space with coefficients in R, a 1960 question of N. E. Steenrod, for a commutative ring R satisfying mild conditions. In the fundamental case R = Z, our result states that the only polynomial cohomology rings over Z which can occur, are tensor products of H ∗ (CP ∞ ; Z) ∼ = Z[x2], H ∗ (B SU(n); Z) ∼ = Z[x4, x6,..., x2n], and H ∗ (B Sp(n); Z) ∼ = Z[x4, x8,..., x4n]. Our classification extends Notbohm’s solution for R = Fp, p odd. Odd degree generators, excluded above, only occur if R is an F2algebra and in that case the recent classification of 2compact groups by the authors can be used instead of the present paper. Our proofs are short, but rely crucially on the general theory of pcompact groups. 1.
AUTOMORPHISMS OF ROOT DATA, MAXIMAL TORUS NORMALIZERS, AND pCOMPACT GROUPS
, 2005
"... Abstract. We describe the outer automorphism group of a compact connected Lie group as a certain subgroup of the outer automorphism group of its maximal torus normalizer, expressed in terms of the associated root datum. The same subgroup can be defined for connected 2compact groups. We use this to ..."
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Abstract. We describe the outer automorphism group of a compact connected Lie group as a certain subgroup of the outer automorphism group of its maximal torus normalizer, expressed in terms of the associated root datum. The same subgroup can be defined for connected 2compact groups. We use this to show that any homomorphism to the outer automorphism group of a pcompact group can be lifted to an action, analogous to a classical theorem of de Siebenthal for compact Lie groups, and we find a candidate formula for the whole space of selfhomotopy equivalences of any connected 2compact group. The results of this paper play a key role in a subsequent paper by the authors where we prove the conjectured classification of 2compact groups and describe their automorphism spaces. 1.